Katz and Sarnak [1]} showed that for any continuous \(-valued central function \) F\( on the symplectic group \) Sp(2g)\(,\)\(\lim _{q \rightarrow \infty } \langle F(\Theta _Q)\rangle = \int _{\mathrm {Sp}(2g)} F(U) dU,\)\(where\)\(\langle F \rangle := \frac{1}{\#\mathcal {H}_{2g+1}} \sum _{Q \in \mathcal {H}_{2g+1}} F(Q).\)\(\)
